Find the values of a and b in a limit

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In summary, the conversation discusses a problem involving a limit and two equations, where one equation involves parameters a and b. The conversation suggests using Taylor's theorem to express f(h) and f(2h) as linear polynomials in h plus an error term that decreases superlinearly. However, it is mentioned that this approach may not be expected given the definition of derivative previously discussed. Other methods, such as decomposing the limit into derivatives of f, are suggested.
  • #1
songoku
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Homework Statement
Please see below
Relevant Equations
Limit

L'Hopital Rule is not allowed
1664334550562.png


I know $$\lim_{h\rightarrow 0} af(h)+bf(2h)−f(0)=0$$
$$a+b=1$$

But I don't know how to find the second equation involving a and b. I imagine I need to somehow obtain ##h## in numerator so I can cross out with ##h## in denominator but I don't have idea how to get ##h## in the numerator.

Thanks
 
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  • #2
Try using Taylor's theorem to express ##f(h)## as a linear polynomial in ##h## plus an error term that decreases superlinearly, and do the same for ##f(2h)##.
 
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  • #3
Office_Shredder said:
Try using Taylor's theorem to express ##f(h)## as a linear polynomial in ##h## plus an error term that decreases superlinearly, and do the same for ##f(2h)##.
Based on the definition of derivative that he gave in another thread, I think this is not the approach that is expected.
 
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  • #4
FactChecker said:
Based on the definition of derivative that he gave in another thread, I think this is not the approach that is expected.
Yes, my lesson has not covered Taylor's theorem

Office_Shredder said:
Try using Taylor's theorem to express ##f(h)## as a linear polynomial in ##h## plus an error term that decreases superlinearly, and do the same for ##f(2h)##.
I tried using Taylor's theorem, as suggested, as best as I can and by only taking the term up until ##h## I get ##a+2b=0##

Is there other possible method to solve the question without using Taylor?

Thanks
 
  • #5
songoku said:
Yes, my lesson has not covered Taylor's theoremI tried using Taylor's theorem, as suggested, as best as I can and by only taking the term up until ##h## I get ##a+2b=0##

Is there other possible method to solve the question without using Taylor?

Thanks
Why not do the obvious decomposition of the limit into derivatives of ##f##?
 
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  • #6
PeroK said:
Why not do the obvious decomposition of the limit into derivatives of ##f##?
I think I understand this hint.

Thank you very much for the help Office_Shredder, FactChecker, PeroK
 
  • #7
songoku said:
PeroK said:
Why not do the obvious decomposition of the limit into derivatives of ##f##?

I think I understand this hint.
Just in case you don't quite understand that hint, or the advice from @FactChecker in Post #3 ...

You had the following as one of the definitions for ##f'(x_0)##.

##\displaystyle f'(x_0)=\lim_{h \to 0} \frac{f(x_0+h)-f(x_0)}{h} ##

So that ##\displaystyle \ f'(0)=\lim_{h \to 0} \frac{f(h)-f(0)}{h} \ .##

You may not realize that ##\displaystyle \ \lim_{h \to 0} \frac{f(2h)-f(0)}{2h} = f'(0)\ ## as well.

(Added a short time later with the Edit feature):

In decomposing the given limit, take the ##\displaystyle \frac{b\,f(2h)}{h} ## term, for instance and write it as ##\displaystyle \frac{b\,(f(2h)-f(0))}{h} + \frac{b\,(f(0))}{h} \ .##

Do similarly with ##\displaystyle \frac{a\,f(h)}{h}##, combine like terms, etc.
 
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FAQ: Find the values of a and b in a limit

What is a limit?

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a specific value. It is used to determine the value that a function approaches as its input gets closer and closer to a particular value.

How do you find the values of a and b in a limit?

To find the values of a and b in a limit, you can use algebraic techniques such as factoring, simplifying, and manipulating equations. You can also use graphical methods, such as plotting the function and visually determining the values of a and b.

What is the importance of finding the values of a and b in a limit?

Finding the values of a and b in a limit is important because it helps us understand the behavior of a function and its relationship to its input. It also allows us to make predictions about the behavior of a function at a certain point and to solve more complex problems in calculus and other areas of mathematics.

Are there any specific techniques for finding the values of a and b in a limit?

Yes, there are specific techniques for finding the values of a and b in a limit, such as the squeeze theorem, L'Hôpital's rule, and the substitution method. These techniques can be used to solve different types of limits, depending on the form of the function.

What are some common mistakes when finding the values of a and b in a limit?

Some common mistakes when finding the values of a and b in a limit include not considering the domain of the function, not simplifying the expression enough, and not using the correct technique for the type of limit. It is also important to check for any discontinuities or points of discontinuity in the function before evaluating the limit.

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